Order Parameters and Color-Flavor Center Symmetry in QCD

Common lore suggests that N-color QCD with massive quarks has no useful order parameters that can be nontrivial at zero baryon density. However, such order parameters do exist when there are n₅ quark flavors with a common mass and d≡gcd (n₅, N) >1. These theories have a Z₃ color-flavor center symmetry arising from intertwined color center transformations and cyclic flavor permutations. The symmetry realization depends on the temperature, baryon chemical potential, and value of n₅/N, with implications for conformal window studies and dense quark matter.